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Advances in Calculus of Variations

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Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

Adolfo Arroyo-RabasaORCID iD: http://orcid.org/0000-0002-8329-1506 / Guido De Philippis / Filip RindlerORCID iD: http://orcid.org/0000-0003-2126-3865
Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/acv-2017-0003


We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

Keywords: Lower semicontinuity; functional on measures; generalized Young measure

MSC 2010: 49J45; 35J50; 28B05; 49Q20; 74B05


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About the article

Received: 2017-01-27

Revised: 2017-10-04

Accepted: 2017-12-20

Published Online: 2018-01-11

Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: EP/L018934/1

Adolfo Arroyo-Rabasa is supported by a scholarship from the Hausdorff Center of Mathematics and the University of Bonn; the research conducted in this paper forms part of his Ph.D. thesis at the University of Bonn. Guido De Philippis is supported by the MIUR SIR-grant “Geometric Variational Problems” (RBSI14RVEZ). Filip Rindler acknowledges the support from an EPSRC Research Fellowship on “Singularities in Nonlinear PDEs” (EP/L018934/1).

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2017-0003.

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